Polyalphabetic cipher machines and rotors

The Jefferson cylinder is a deceptively simple device which implements a polyalphabetic
substitution cipher; conceived in the late 18th century, it had remarkable cryptographic
strength for its time. Polyalphabetic substitution ciphers implemented by a class of rotor based machines were the dominant cryptographic tool in World War II. Such machines, including the Enigma machine and those of Hagelin, have an alphabet which changes continuously for a very long period before repeating; this provides protection against Kasiski analysis and methods based on the index of coincidence.

Jefferson cylinder

The Jefferson cylinder implements a polyalphabetic substitution cipher while avoiding complex machinery, extensive user computations, and Vigen'ere tableaus. A solid
cylinder 6 inches long is sliced into 36 disks. A rod inserted through the cylinder axis allows the disks to rotate. The periphery of each disk is divided into 26 parts. On each disk, the letters A-Z are inscribed in a (different) random ordering. Plain text messages are encrypted in 36-character blocks. A reference bar is placed along the cylinder's length. Each of the 36 wheels is individually rotated to bring the appropriate character (matching the plaintext block) into position along the reference line. The 25 other parallel reference positions then each define a ciphertext, from which (in an early instance of randomized encryption) one is selected as the ciphertext to transmit.

The second party possesses a cylinder with identically marked and ordered disks (1-
36). The ciphertext is decrypted by rotating each of the 36 disks to obtain characters along a fixed reference line matching the ciphertext. The other 25 reference positions are examined for a recognizable plaintext. If the original message is not recognizable (e.g., random data), both parties agree beforehand on an index 1 through 25 specifying the offset between plaintext and ciphertext lines.

To accommodate plaintext digits 0-9 without extra disk sections, each digit is permanently assigned to one of 10 letters (a,e,i,o,u,y and f,l,r,s) which is encrypted as above but annotated with an overhead dot, identifying that the procedure must be reversed. Reordering disks (1 through 36) alters the polyalphabetic substitution key. The number of possible orderings is 36!=3.72 × 1041. Changing the ordering of letters on each disk affords 25! further mappings (per disk), but is more difficult in practice.

Rotor-based machines - technical overview

A simplified generic rotor machine consists of a number of rotors (wired code wheels) each implementing a different fixed mono-alphabetic substitution, mapping a character
at its input face to one on its output face. A plaintext character input to the first rotor
generates an output which is input to the second rotor, and so on, until the final ciphertext
character emerges from the last. For fixed rotor positions, the bank of rotors collectively
implements a mono-alphabetic substitution which is the composition of the substitutions
defined by the individual rotors.

To provide polyalphabetic substitution, the encipherment of each plaintext character
causes various rotors to move. The simplest case is an odometer-like movement, with a
single rotor stepped until it completes a full revolution, at which time it steps the adjacent
rotor one position, and so on. Stepping a rotor changes the mono-alphabetic substitution
it defines (the active mapping). More precisely, each rotor Ri effects a mono-alphabetic
substitution fi. Ri can rotate into ti positions (e.g., ti = 26). When offset j places from a reference setting, Ri maps input a to fi(a-j)+j, where both the input to fi and the final output are reduced mod 26.

The cipher key is defined by the mono-alphabetic substitutions determined by the fixed
wheel wirings and initial rotor positions. Re-arranging the order of rotors provides additional variability. Providing a machine with more rotors than necessary for operation at any one time allows further keying variation (by changing the active rotors).

Fact: Two properties of rotor machines desirable for security-related reasons are: (1) long periods; and (2) state changes which are almost all "large". The second property concerns the motion of rotors relative to each other, so that the sub-mappings between rotor faces change when the state changes. Rotor machines with odometer-like state changes fail to achieve this second property.

Rotor machine output methods: Rotor machines were categorized by their method of
providing ciphertext output. In indicating machines, ciphertext output characters are indicated by means such as lighted lamps or displayed characters in output apertures. In printing machines, ciphertext is printed or typewritten onto an output medium such as paper. With on-line machines, output characters are produced in electronic form suitable for direct transmission over telecommunications media.